A Hydrogen Atom Initially in the Ground Level Absorbs a Photon, Which Excites It to the N = 4 Level. Determine the Wavelength and Frequency of the Photon. - Physics

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A hydrogen atom initially in the ground level absorbs a photon, which excites it to the n = 4 level. Determine the wavelength and frequency of the photon.

A hydrogen atom initially in the ground state absorbs a photon which excites it to the n = 4 level. Estimate the frequency of the photon

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Solution

For ground level, n1 = 1

Let E1 be the energy of this level. It is known that E1 is related with n1 as:

`"E"_1 = (-13.6)/("n"_1^2) "eV"`

`= (-13.6)/1^2`

= −13.6 eV

The atom is excited to a higher level, n2 = 4.

Let E2 be the energy of this level.

∴ `"E"_2 = (-13.6)/"n"_2^2 "eV"`

= `(-13.6)/4^2`

= `(-13.6)/16 "eV"`

The amount of energy absorbed by the photon is given as:

E = E2 − E1

= `(-13.6)/16 - ((-13.6)/1)`

= `(13.6 xx 15)/16 "eV"`

= `(13.6 xx 15)/16 xx 1.6 xx 10^(-19)`

= 2.04 × 10−18 J

For a photon of wavelength λ, the expression of energy is written as:

`"E" = "hc"/lambda`

Where,

h = Planck’s constant = 6.6 × 10−34 Js

c = Speed of light = 3 × 108 m/s

∴ λ = `"hc"/"E"`

= `(6.6 xx 10^-34 xx 3 xx 10^8)/(2.04 xx 10^-18)`

= 9.7 × 10−8 m

= 97 nm

And, frequency of a photon is given by the relation,

`"v" = "c"/lambda`

= `(3 xx 10^8)/(9.7 xx 10^(-8))`

≈ 3.1 × 1015 Hz

Hence, the wavelength of the photon is 97 nm while the frequency is 3.1 × 1015 Hz.

Concept: Energy Levels
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Chapter 12: Atoms - Exercise [Page 436]

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